(3) (Adopted from Gottfried) As we will see in future quantum mechanics courses, the
operators that we study in this problem play an essential role in the theory of many-fermion
systems. They allow us to describe states that conform to the Pauli Principle.
The Hamiltonian of a system is given by
H=
k
X
n,m=1
a†
nAnmam,
where Anm are the elements of a Hermitian matrix with eigenvalues E1,E2,···,Ek. The a’s
are operators and satisfy
{am, an}= 0
{a†
m, a†
n}= 0 (1)
{a†
m, an}=δm,n
1. Show that it is possible to introduce a new set of operators
αn=
k
X
m=1
Unmam
that also satisfy the commutation relations in Eq. (1), but in terms of which
H=
k
X
n=1
Enα†
nαn,
and where the Unm are elements of a unitary matrix.
2. Show that the operators Nn=α†
nαn(with n= 1,2,···, k) are compatible.
3. Let νnbe the eigenvalues of Nn. Show that νn= 0 or 1.
4. Let |νni(i.e., |0ior |1i) be an eigenket of Nn. Show that
αn|0i= 0;
α†
n|1i= 0;
α†
n|0i=eiθ|1i,
where θis real.
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